The Bayesian methods for introducing sparsity to the covariance matrix have been also developed. The G-inverse Wishart prior, considered by [1]}, was often used in the Bayesian framework. However, the method had a limitation in applications as the dimension gets higher because of posterior intractability. [2]} extended G-inverse Wishart prior to a broader class and proposed a blocked Gibbs sampler to sample covariance matrices from the resulting posterior, but [2]} considered only decomposable graphs. [4]} also considered covariance graph model for the inference of covariance matrices and also provided a blocked Gibbs sampler that is applicable to all graphs but under the continuous spike and slab prior, which uses the mixture of two Gaussian distributions on off-diagonal entries, one with sufficiently small variance and the other with variance substantially far from 0, and the exponential prior on diagonal entries. But the method proposed by [4]} had inherent difficulty in introducing sparsity to the structure of the model due to the absolute continuity of prior. Furthermore, none of [1]}, [2]}, and [4]} did not attain any asymptotic statistical properties. [9]} were able to introduce sparsity to the covariance structure using beta-mixture shrinkage prior and attained the posterior convergence rate of covariance matrix under the Frobenius norm. However, compared to frequentist literature, Bayesian literature still lacks in the methods for inference of sparse covariance matrices, and most of the proposed methods are not theoretically well supported. Furthermore, they have a limit in introducing sparsity to the covariance structures due to the absolute continuity of prior.
